Foci of an Ellipse

Date: 06/08/2001 at 13:52:00
From: Scott Nugent
Subject: Elliptical foci
If the major and minor axis of an ellipse are given, how do I find the
focus points?

Date: 06/08/2001 at 15:40:43
From: Doctor Floor
Subject: Re: Elliptical foci
Hi, Scott,
Thanks for writing.
The foci F1 and F2 of an ellipse lie on its major axis, at equal
distances from the center M of the ellipse. Let P be a variable point
on the ellipse; then the sum of distances d(P,F1)+d(P,F2) is constant.
Now let the major axis meet the ellipse in X1 and X2 and the minor
axis in Y1 and Y2. Let a be the length of the major axis. In a figure:
Y1
|
|
X1--F1----M-----F2--X2
|
|
Y2
Clearly
d(X1,F1)+d(X1,F2) = d(X1,F1)+d(X2,F1) = a
and thus d(Y1,F1) = d(Y2,F2) = a/2.
Knowing this it is not too difficult to find the position of F1 and F2
(for instance we might use Pythagoras' theorem in triangle F1MY1 to
compute the distance from M to F1).
If you need more help, just write back.
Best regards,
- Doctor Floor, The Math Forum
http://mathforum.org/dr.math/